In conventional and high transition temperature copper oxide and iron pnictide superconductors, the Cooper pairs all have even parity. As a rare exception, Sr2RuO4 is the first prime candidate for topological chiral p-wave superconductivity, which has time-reversal breaking odd-parity Cooper pairs known to exist before only in the neutral superfluid 3 He. However, there are several key unresolved issues hampering the microscopic description of the unconventional superconductivity. Spin fluctuations at both large and small wavevectors are present in experiments, but how they arise and drive superconductivity is not yet clear. Spontaneous edge current is expected but not observed conclusively. Specific experiments point to highly band-and/or momentum-dependent energy gaps for quasiparticle excitations in the superconducting state. Here, by comprehensive functional renormalization group calculations with all relevant bands, we disentangle the various competing possibilities. In particular we show the small wavevector spin fluctuations, driven by a single two-dimensional band, trigger p-wave superconductivity with quasi-nodal energy gaps.PACS numbers: 74.20.Rp, 71.27.+a Very soon after the discovery of superconductivity in Sr 2 RuO 4 [1], it was proposed that the superconducting (SC) pairing is of unconventional nature [2,3]. Later experiments have provided evidence that the Cooper pair in the SC state is of odd parity [4] with total spin equal to one [5]. Further evidence indicates the superconductivity to be chiral, breaking time reversal symmetry [6,7]. Sr 2 RuO 4 is thus the first prime candidate for a chiral p-wave superconductor [8][9][10][11], an interesting analogue of the neutral superfluid 3 He. It has recently received great interest as by suitable manipulations it may support zero energy Majorana bound states in vortices [12], the building block for topological quantum computing [13]. However, there are a number of outstanding issues associated with the chiral p-wave superconductivity in Sr 2 RuO 4 . First, p-wave spin triplet pairing is expected to be associated with spin fluctuations at small wavevector. However, the spin density wave (SDW) fluctuation observed in Sr 2 RuO 4 is dominated by a large wavevector at higher temperatures and coexist with a feature at small wavevector at lower temperatures. [14] A resolution of this puzzle is vital to understand the superconductivity. Second, one would expect a spontaneous electric current at the edge of the RuO 2 layers as a result of the chiral SC state. The edge current, however, has not been observed conclusively in experiments.[15] One possible reason is the edge current is very fragile and difficult to establish against disorders. Another possibility is a topological cancellation from hole-like and electron-like bands, [16] posing a question as whether the SC state is topologically nontrivial at all. Third, the specific measurement reveals abundance of low energy quasiparticle excitations below the transition temperature. [17] This would point to mu...